MAP Estimation, Message Passing, and Perfect Graphs

TL;DR

This paper extends the class of graphs where MAP estimation can be efficiently solved exactly using message passing algorithms, by leveraging perfect graph theory and linear programming relaxations.

Contribution

It introduces a framework connecting perfect graphs with MAP estimation, enabling polynomial-time solutions and exact message passing recovery for broader graph classes.

Findings

MAP estimation is in P for perfect graphs.

Message passing algorithms recover exact solutions on perfect graphs.

Linear programming relaxations are integral for perfect graphs.

Abstract

Efficiently finding the maximum a posteriori (MAP) configuration of a graphical model is an important problem which is often implemented using message passing algorithms. The optimality of such algorithms is only well established for singly-connected graphs and other limited settings. This article extends the set of graphs where MAP estimation is in P and where message passing recovers the exact solution to so-called perfect graphs. This result leverages recent progress in defining perfect graphs (the strong perfect graph theorem), linear programming relaxations of MAP estimation and recent convergent message passing schemes. The article converts graphical models into nand Markov random fields which are straightforward to relax into linear programs. Therein, integrality can be established in general by testing for graph perfection. This perfection test is performed efficiently using a polynomial time algorithm. Alternatively, known decomposition tools from perfect graph theory may be used to prove perfection for certain families of graphs. Thus, a general graph framework is provided for determining when MAP estimation in any graphical model is in P, has an integral linear programming relaxation and is exactly recoverable by message passing.